Reduced-Order Modeling for Heston Stochastic Volatility Model

Year 2024, Early Access, 1 - 14

Sinem Kozpınar , Murat Uzunca , Bülent Karasözen

https://doi.org/10.15672/hujms.1066143

Abstract

In this paper, we compare the intrusive proper orthogonal decomposition (POD) with Galerkin projection and the data-driven dynamic mode decomposition (DMD), for Heston's option pricing model. The full order model is obtained by discontinuous Galerkin discretization in space and backward Euler in time. Numerical results for butterfly spread, European and digital call options reveal that in general DMD requires more modes than the POD modes for the same level of accuracy. However, the speed-up factors are much higher for DMD than POD due to the non-intrusive nature of the DMD.

Keywords

option pricing, Heston model, discontinuous Galerkin method, reduced order modeling, proper orthogonal decomposition, dynamic mode decomposition

References

• L. W. Ballestra, G. Pacelli. Pricing European and American options with two stochastic factors: A highly efficient radial basis function approach. Journal of Economic Dynamics and Control, 37(6):1142 – 1167, 2013.
• M. Balajewicz, J. Toivanen. Reduced Order Models for Pricing American Options under Stochastic Volatility and Jump-diffusion Models Procedia Computer Science , 80:734 – 743, 2016.
• Marko Budisic. Matlab toolbox for Koopman mode decomposition. Technical report, Department of Mathematics, University of Wisconsin - Madison, 2015.
• O. Burkovska, B. Haasdonk, J. Salomon, B. Wohlmuth. Reduced basis methods for pricing options with the Black–Scholes and Heston models. SIAM Journal on Financial Mathematics, 6(1):685–712, 2015.
• K. K. Chen, J.H. Tu, C.W. Rowley. Variants of dynamic mode decomposition: Boundary condition, Koopman, and Fourier analyses. Journal of Nonlinear Science, 22(6):887–915, 2012.
• R. Cont, N. Lantos, O. Pironneau. A reduced basis for option pricing. SIAM J. Fin. Math., 2(1):287–316, 2011.
• L.X. Cui, C. Long. Trading strategy based on dynamic mode decomposition: Tested in Chinese stock market. Physica A: Statistical Mechanics and its Applications, 461:498 – 508, 2016.
• B. Düring, M. Fournié, C. Heuer. High-order compact finite difference schemes for option pricing in stochastic volatility models on non-uniform grids. Journal of Computational and Applied Mathematics, 271:247 – 266, 2014.
• R. England. The use of numerical methods in solving pricing problems for exotic financial derivatives with a stochastic volatility. PhD thesis, M. Sc. Thesis, University of Reading, 2006.
• S. L. Heston. A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies, 6(2):327–343, 1993.
• K. J. In’T Hout, S. Foulon. ADI finite difference schemes for option pricing in the Heston model with correlation. International Journal of Numerical Analysis and Modeling, 7(2):303–320, 2010.
• M. R. Jovanovi’c, P. J. Schmid, J.W. Nichols. Sparsity-promoting dynamic mode decomposition. Physics of Fluids, 26(2), 2014.
• B. O Koopman. Hamiltonian systems and transformation in Hilbert space. Proceedings of the National Academy of Sciences, 17(5):315–318, 1931.
• S. Kozpınar, M. Uzunca, and B. Karasözen. Option pricing under Heston stochastic volatility model using discontinuous Galerkin finite elements. Mathematics and Computers in Simulation , 177 568-58, 2020.
• K. Kunish, S. Volkwein. Galerkin proper orthogonal decomposition methods for parabolic problems. Numerische Mathematik, 90(1):117–148, 2001.
• V. L. Lazar. Pricing digital call option in the Heston stochastic volatility model. Studia Univ. Babeş-Bolyai Math., 48(3):83–92, 2003.
• A. Lipton. Mathematical methods for foreign exchange: A financial engineer’s approach. World Scientific, 2001.
• J. Mann and J.N. Kutz. Dynamic mode decomposition for financial trading strategies. Quantitative Finance, 16(11):1643–1655, 2016.
• A. Mayerhofen, K. Urban. A reduced basis method for parabolic partial differential equations with parameter functions and applications to option pricing. Journal of Computational Finance 20(4): 71-106, 2016.
• B. Peherstorfer, P. Gómez, H. M. Bungartz. Reduced models for sparse grid discretizations of the multi-asset Black-Scholes equation. Advances in Computational Mathematics, 41(5):1365–1389, 2015.
• O. Pironneau. Pricing futures by deterministic methods. Acta Numerica, 21:577–671, 2012.
• B. Rivière. Discontinuous Galerkin methods for solving elliptic and parabolic equations, Theory and implementation. SIAM, 2008.
• E. W. Sachs, M. Schu. A priori error estimates for reduced order models in finance. ESAIM:Mathematical Modelling and Numerical Analysis, 47:449–469, 3 2013.
• P. J. Schmid. Dynamic mode decomposition of numerical and experimental data. Journal of Fluid Mechanics, 656:5–28, 2010.
• D.Y. Tangman, A. Gopaul, M. Bhuruth. Numerical pricing of options using high-order compact finite difference schemes. Journal of Computational and Applied Mathematics, 218(2):270–280, 2008.
• J.H . Tu, C.W Rowley, D.M Luchtenburg, S. L Brunton, J.N. Kutz. On dynamic mode decomposition: theory and applications. J. Comput. Dyn., 1(2):391–421, 2014.
• S. Volkwein. Model Reduction using Proper Orthogonal Decomposition. Lecture Notes, University of Konstanz, 2013.
• G. Winkler, T. Apel, U.Wystup. Valuation of options in Heston’s stochastic volatility model using finite element methods. Foreign Exchange Risk, pages 283–303, 2001